What is "principal ideal"

n. (context algebra English) An ideal which is generated by a single element of the ring.

In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P.

The remainder of this article addresses the ring-theoretic concept.